We hope that Pierre Leroux would have enjoyed this tribute to him. Not only was
he a great "pure" combinatorialist, one of the co-founders of the very abstract
theory of species, he was also very interested in symbolic computation,
and inspired his students to concretize these abstract species by programming them.
While the present article does not talk about species, we are sure that
Pierre would have liked it all the same, especially since it tackles a conjecture
discussed by Richard Stanley, in the problem session that he chaired, back in May 1985,
in the conference so efficiently and warmly organized by him and Gilbert Labelle.

Added Feb. 19, 2009: read the abstract of a
talk
I gave about this work at the Rutgers Experimental Math seminar.
Important: This article is accompanied by the following two
Maple packages

qTSPP,
that contains the substance of our result, a q-holonomic description
of the function B(n,j) of the paper. It is called qB(n,j) in the package.
It also contains the procedure CheckqTSPP(N), that empirically verifies
identities (Soichi) and (Okada) from the paper for all n &le N,
thereby supplying a semi-rigorous proof (the larger N, the better).

Sample Input and Output

Once you have the Maple package
qTSPP,
and want to emprically check (Soichi) and (Okada) for n ≤ 100,
unfortunately, at present (at least on our small computer) you can't
use symbolic q, but you can plug-in any numeric value of q and see
if you get it right.

Once you have the Maple package
TSPP,
as well as the input file
inTSPP400,
both in the same directory, staying in that directory,
in order to verify (Soichi) and (Okada) for n ≤ 400
doing
maple -q < inTSPP400 > oTSPP400
would yield the
output.

Added Aug. 6, 2008:
We have found a j-free operator ,of the form P(N,J,n), where N and J are
the fundamental shift operators in the n and j variables respectively,
annihilating B(n,j) (for the q=1 case).
This is very encouraging, since it indicates that with a bit of more
computation, one should get a j-free operator annihilating
the summand of (Soichi) and (Okada) enabling a fully rigorous proof
in the style of Sister Celine.

This operator has been added to
TSPP,
as procedure ManueljFree(n,j); . Procedure CheckManueljFree verifies
it empirically for all 1<=jTSPP,
as well as the input file
inMa500,
both in the same directory, staying in that directory,
in order to check that the proposed j-free operator does indeed annihilate
B(n,j) for 1 ≤ j ≤ n ≤ 500
doing
maple -q < inMa500 > oMa500
would yield the
output.
Added June 5, 2009: Christoph Koutschan just won a $300 prize (from my personal funds) that I offered him for
extending the method of the present article to get a
fully rigorous proof of
the q=1 case, i.e. he (and his computer) gave a completely "human-free" proof of Stembridge's so-called TSPP theorem
(or rather of the Okada determinant evaluation that was shown by HB (Human Being) Soichi Okada
to imply it).
Congratulations, Christoph!, and good luck in doing the q-case, and making narrow-minded pedants
like the editors of SLC "happy" (aha, of course, they won't be happy, they are happier now that they can
claim that we don't "realy" have a proof).
Doron Zeilberger's List of Papers